AFCRC TN-59-953
/O^. ^* -r^ NEW YORK UNIVERSITY
m
Institute of Mathematical Sciences ^^ v/averiy j-^^:,, Now YoHc 3, n. y.
Division of Electromagnetic Research
RESEARCH REPORT No. EM-147
Decay Exponents and Diffraction Coefficients for
Surface Waves on Surfaces of Non-Constant Curvature
JOSEPH B. KELLER and BERTRAM R. LEVY
Contract No. AF 19(604)5238
OCTOBER 1959
si ;j
AFCRC-TN-$9-953
NEW YORK UNIVERSITY
Institute of Mathematical Sciences
Division of Electromagnetic Research
Research Report No. EM-1^7
DECAY EXPONENTS AND DIFFRACTION COEFFICIENTS FOR
SURFACE WAVES ON SURFACES OF NON- CONSTANT
CURVATURE
by
Joseph B. Keller
and
Bertram R. Levy
fAJLKd^
JosejM B. Keller
Bertram R . LeAry
rrls Kline Dr. Werner Gerbes fJ
Morris Kline
Project Director Contract Monitor
The research reported in this document has
been sponsored by the Air Force Cambridge
Research Center, Air Research and Develop-
ment Command, \mder Contract No. AF 19(60^)5238.
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Abstract
The diffraction of a plane scalar wave by a hard elliptic cylinder is
investigated theoretically. The field is obtained and expanded asymptotically
for incident wavelengths small compared with the dimensions of the generating
ellipse. The method of obtaining the asymptotic expansion of the diffracted
field parallels the methods of reference [9] • However^ additional term^ in
the asymptotic expansion are obtained. In reference [9] it was shown that
the asymptotic expansion of the diffracted field was in agreement with the
geometrical theory of diffraction as presented in references [h^ and [5] .
The additional terms in the field obtained in this paper we Interpret geo-
metrically as higher order corrections to the decay exponents and diffraction
coefficients as given in reference [5] . Finally,, we obtain additional terms
to those given in reference [8] for the asymptotic expansion of the field
diffracted by a parabolic cylinder. We then show that these higher order
corrections have the same geometrical interpretation as in the case of the
elliptic cylinder. The determination of these corrections permits the
geometric theory to be extended to longer wavelengths than could be treat-
ed previously. Similar results are obtained for soft cylinders. Then the
field on a hard convex cylinder of arbitrary shape is determined asymptotically
by a quite different method - that of asymptotically solving an Integral
equation. The result is found to coincide with the generalization based
upon the solution for the elliptic cylinder.
Table of Contents
Page
1. Introduction 1
2. Diffraction by an elliptic cylinder 3
3. Diffraction by a parabolic cylinder I3
h-. Integral equation method 21
Appendix 32
References 35
- 1 -
1. Introduction
When a wave is incident upon an opaque object large compared to the
incident wavelength a shadow is iTonned. Some radiation penetrates into
the shadow. The first quantitative analysis of this penetration effect
for the case of a smooth object was that of G. N. Watson ^ -^ . He showed
that the field in the shadow of a sphere consists of a sum of modes.
Each mode decays exponentially with increasing distance from the shadow
boundary into the shadow. Numerous authors have pursued Watson's analysis,
considering spheres which are not opaque or which are surrounded by non-uniform
[21
media. Many of these investigator are described by H. Bremmer ^ -^ . Independent-
ly W. Franz and K. Depperman ^ -^ discovered the existence of an exponentially
decaying wave travelling around a circular cylinder. They also observed that
this wave continues travelling into the illuminated region. These results,
as well as those referred to above pertain to bodies of constant curvature.
What are the corresponding results for objects of non-constant ciirvature?
This question is answered by the geometrical theory of dlffration Introduced
by J . B. Keller'--', which predicted that radiation travels along surface rays.
These rays are geodesies on the surface of any object which originate at the
shadow boundary. They continually shed diffracted rays which Irradiate the
shadow and also enter the illuminated region. A quantitative theory of the
field diffracted by a cylinder of arbitrary convex cross-section was constructed
[51
with the aid of these rays '- -^ . In this theory certain decay exponents and
diffraction coefficients were introduced. The decay exponents detennine the
rate of decay of the various field modes along a surface ray.. The diffraction
coefficients determine the amplitudes of the various modes on a surface ray,
and the amplitude of the field on the shed diffracted rays. It was assumed
that the decay exponents and diffraction coefficients depend only upon local
properties of the ray and the surface. By comparing the predictions of this
theory with the results of W. Franz'- -' for the circular cylinder, the decay
exponents and diffraction coefficients were determined. A similar analysis
was performed for three dimensional curved objects by B. R. Levy and
J . B . Keller ^^-^ .
The results of the geometrical theory of diffraction have been tested
by comparing them with the exact solutions of certain diffraction problems
involving objects of non-constant curvature. To make this comparison it was
necessary to expand the exact solution asymptotically for wavelength small
compared to the dimensions of the object. This has been done for the field
diffracted by a parabolic cylinder by S. 0. Rice ^ -^ , an elliptic cylinder
by B. R. Levy'-^J and by R. K. Ritt and N. D. Kazarinoff L J, and for an
ellipsoid of revolution by J. B. Keller and B. R. Levy I- -I and by R . K. Ritt
ri2i
and N. D. Kazarinof f l- -^ . In all cases the leading term in the asymptotic
expansion agreed precisely with the results of the geometrical theory.
We now propose to improve the geometrical theory of diffraction by an
arbitrary cylinder so that it will also yield the next term in the asymptotic
expansion. To this end we must determine the next terms in the expressions for
the decay exponents and the diffraction coefficients. The previously determined
terms involve the radius of curvature of the cylinder. The new terms will
Involve the derivative of the radius with respect to arclength along the cross-
sectional curve. To find the new terms we shall examine the next term in the
asymptotic expansion of the exact expression for the field diffracted by an
elliptic cylinder. We shall express terms of local geometrical quantities
such as the radius of curvature and its derivative. Then we shall assimie that
- 3 -
the final geometrical expression is correct for an arbitrary cylinder. As a
first test of this result, we shall show that It correctly yields the next
term for the field diffracted by a parabolic cylinder. Of course, it also
yields the correct term in the case of a circular cylinder. The results
are also obtained by asymptotically solving the integral equation for the
cylinder current . These results coincide with those obtained by general-
izing the results obtained for the elliptic cylinder.
The determination of these new corrections permits us to use our theory
for longer wavelengths than could have been treated previously . The improve-
ment resulting from the correction to the decay exponent is shown in \j] .
2. Diffraction by an elliptic cylinder
Let us consider the field u produced by a line source parallel to the
generators of an elliptic cylinder. Then u is the solution of the following
problem
(A + k^)u = 5(1-1^) 6(n) (1)
Ma^ =0 (2)
lim r(iku - u ) = . (3]
r — >oo
For simplicity the source has been taken to lie in the plane containing the
major axis of the ellipse. The elliptic coordinates (|,r|) are related to
cartesian coordinates by the equations
X = h cosh ^ cos T\ (k)
y = h slnh t sin t) (5)
k -
In (k) and (5) h denotes one half the interfocal distance of the ellipses
I = constant^ of which | = a is the cross-section of the cylinder.
In reference [9] it is shown that on the cylinder the solution of
(1) - (5) can he written in the form
u(a,Ti) = (kh) Yl ^^ r '(^) TTT'
n=l "" ^n ^""^ h/^-b V ^^^ (a)
d|2
For large kh it has the asymptotic expansion
(6)
The functions C and V are defined and asymptotically expanded for
large kh in reference [9] . A breif review of the pertinent properties
of these functions follows.
The function V is the outgoing solution of the equation
^"-^'^ 2 2 2 (1)
(kh)'' (b/ - cosh'^OV^^^'' = . (7)
V ^"-ht)^ ^^/^ 3^/5 ^-1 (, 2 _ ,^3^2^)-iA A(3i/53-W3(^)2/3^^ ^Q^
n
Here b is defined by
n
vj^^'(a) = . (9)
The functions C and A are defined by
I
I ^5/^ = - I (b^^ - cosh^x)^/^ dx (10)
cosh
■h
- 5
A(t) = I cos(z'^ - tz)dz . (11)
The function C (t]) is the even solution of the equation
c" + (kh)^(b ^ - cos^Ti)C = . (12)
For large kh it has the asymptotic expansion
C^^ cos[kh j (b/ - cos2Ti)l/2dT[] |(b/ - l)/(b^^ - cos^Ti)! .(13)
We now specialize (6) to the case of plane vave incidence. To do this
we miatiply (6) by e^'^^' 2^/'^7r''"' ^(kh cosh i )"'-/^exp [-ikh cosh t ] and let
E -> oo . Then we obtain
o
CD b C (,-«) e^PC-ikh j (b^2 _ ,„32^)l/2^^2
u(a,n) = E y: ^V^ ° — nv • ^1"^)
n=l C (rt) h/b-b V^^' (a)
Here E = e ' 2 ' (kh) ' . Upon expanding the C function we find that
{l^) becomes
u(a,Ti)- kh ^ ^n y^u "^^^^n "^°^ '^'
(15)
exp[ikhG(jt/2,Ti)] + exp[ikhG(Ti,5jt/2)] j" yl
X rrpn Jl-exp[2ikhG(0,Jt)]
The details of the evaluation of the limit are to be found in reference pQ^
p. l4.
- 6 -
In (15) G Is defined by
G(a,p) = j (b ^ - cos^n)-'-/^ dri
In reference [9] the leading term In the asymptotic expansion of
each of the summands In (15) was computed. In order to carry out this
calculation It was found necessary to compute two termis In the asymptotic
expansion of the eigenvalue b . We shall now compute a further term in
the asymptotic expansion of each of the terms in (l5)- In order to do
this we shall first compute another term in the asymptotic expansion of b .
To do this ue first observe that the leading terms in the asymptotic
expansion of V^ (l) are obtained by differentiating (8). The leading
(1) '
term in the asymptotic expansion of V (|) comes from differentiating
the Airy function A. Therefore ^ will be nearly equal to the result
obtained In [9] so we write it in the form
T
Here t, = ^(a), q is the nth root of the equation A ('l) = 0, and 6
is an as yet undetermined correction which is small compared to unity.
We now set b = cosh a + e and insert this expression into (lO)
n n
which determines ^ . In this way we obtain
2 . 3/2 _ 22/1 (cos.a)V^ ^ 5/2 . ^'\'^'('^^'^'^*1) ^ y,
3 ^n 5 sinh an ^^ , TY^r 7^ \^^ ^ ^ ^^°''
-^ ^ 50 sinh a(cosh a) '
Now we Insert (l?) and the above form of b Into (9) and obtain the
following re stilt for 5
n
6 = ^ e^^/^(cosh^a . 1, sinh^a .7) . o((kh)-'^/5) . (19)
"" 80 2^/^(sinh a cosh a)^/^(kh)^/^
By comparing (18) and (1T)> and using (19) for 6 . we determine t . Then
b is given by
T (sinh a) ' 1 (cosh a + 7)
"" (cosh a)^/5(kh)^/5 30(sinh a)^/5(cosh a)5/5(kh)^/5
2
(2 cosh a-l) ^/ /, , \-2v
201 (sinh a)^/^(cosh a)'^/^(kh)^/^
In (20), the quantity t is defined in terms of q by
ifl/3
Upon substituting (20) into (16) and asymptotically expanding the
the integral, we obtain
P
ikhG(a,p) = ikh I (cosh^a - cos^ti)^/% + i(kh)-^/5( sinh a cosh af^^ (22)
X J (cosh a - cosh tj) ^ di] + -y- — -j-
I 50(kh)-^'''(cosh a sinh a) ^^
p I
I 2 2-'5/2 2 2 2
X (cosh a - cos i]) ' J(cosh a + 7)(cosh a - cos t])
2 2 I , l(2cosh^a-l)
- 15 sinh a cos t] I dr) -T /^ ^,
20T (kh) '-'fslnh a cosh a) ' ''
X I (cosh^a - cosh^ri)""'-/^dTi + 0((kh)""'-)
In reference [9] it was shown that the first two integrals in
(22) have simple geometric interpretations in terms of the arclength s
along the ellipse. To show this we let s and s be the values of s
corresponding to t] = a and 11 = ^ respectively. Then we find that
the first term on the right of (22) is just ik times the arclength
2
ik I ds . (23)
Similarly, the second term on the right of (22) is
Here b(s) denotes the radius of curvature of the ellipse.
We shall now express the third and fcirrth terms appearing on the
right side of (22) in geometric terms.
The third term can he written as
It
30 k
2 -V3,, 16 ^ 2 ^\.
) ds
The fourth term is equal to
20t k
175
'' .-V5(., I ./ . '-^, .3
The second derivatives in (25) and (26) can be eliminated hy Integrating
by parts . Then (25) and (26), respectively become
±1
Q ^
8 s
W75 ■ 5 ^1/5
1 2 r 2
+
b-^/5 (1 + I b 2) ds .
(27)
20 T k
175
1 s
b-^/^ (2 + ^ b ^) ds
9 £>
(28)
Let us now insert (22) into the expression (I5) for u(a,Ti). In
doing so we shall utilize the geometric forms (23), (?h), (27) and (28)
r
for the integrals in (22). We must also evaluate dV (a)/Sb which we
find, by the methods of reference [9], to be given by
- 10 -
a „', . -1,_ v4/3 1/2, . , N-l/2, - ,1/2 -2ijt/3
•^ V (a)--^ jt (kh) ' 2 ' (sinh a) ' (cosh a) ' e '
at ^n
(29)
When all these expressions are inserted into (15). we finally obtain
the following asymptotic formula for u:
u(a,T])
,1/2
'''°°^'' '\iA I (VK)) -'fxpEi^t^. j P^as] (30)
(cosh a - cos T)) f— _ V.
+ exp
ikt^ +1 p^ds
r + 0(kh)
■2/3
T
X ^ 1 + exp[ikT - P ds]
o
In (30) the distances t and t and the points Q , Q and P are as shown
in figure 1. The distance T is the total arclength of the ellipse.
The quantities p and y are defined by
. ,1/3^-2/3 ^"^n -k/3 f,^ 8 ,- 2,
^ = IT^k /^b + ——73 b / (1+ - bfe^ )
(31)
30 k
20t k
;T73 ^ (2-f ^ b^ )
11
7 = exp
^"/^6^/\,5(P) K ^
iT7v7.(P)
U5 60q^
(32)
We shall now relate (50) to the geometric theory of diffraction presented
in reference [5] • When this theory is applied to the present case it yields
/(P) = 2^/V/\-l/V/6e-Wl2^-l/5(p) f- ^(^) 3^ (p) (53)
n=o
Jb^(Q^) exp[ikt^ + J p^ds] + B^CQ^) expQikt^ + f P^, is] I
J 1 - exp [ikT - p^ ds] I
Here p and B are respectively the decay exponent and diffraction coefficient
of the nth mode. The leading terms p and B in the expressions for these
n n
quantities, as given in reference [5] , are
= IX kl/5 b-2/5
n n
{3h)
B^°(P) - ,5/" 2-lA 6-1/6 ^-1/2 j-^^1/2 ^,^y-l 3l«/2\l/6(p) . (35,
Upon comparing (53) and (50) we see that they are identical provided that
the decay exponent p is given by (51) and the diffraction B (P) by
12 -
B fP) = B °(P) y (P) (36)
The new results (3I) and (36) for p and B agree with the previous re-
sults (3^) and (35) to the lowest order in k . The new value of p is
-2h -3/4
valid to 0(k ''^) and the new value of B to 0(k ' ). Thus they con-
tain corrections to the previous results .
The preceding results pertain to a hard elliptic cylinder - i.e
one on which cki/i^n = 0. We have performed a similar calculation for
a soft elliptic cylinder - i.e. one on which u=o. In this case we
also find corrections to the decay exponents and to the diffraction
coefficients. For the decay exponents we obtain
30 k
Here t = 6 <Le ' and q is the nth root of the equation
A(%,) = 0. For the diffraction coefficients we find
iit/6.1/5, /^N c
; 6 ' t^(P) q^
B (P) = B °(P) exp
45 k^/V/5(P)
(38)
Here B (P) is the lowest order result for the diffraction coefficient,
n ^
given in reference [5] ^ as _
B^°(P) = n^A 2IA 6-2/5 ^-l/l2f-^'(^)-j-l ^i.M ^1/6 (35)
13
The new results (37) and (39) contain corrections to the previous
results, as in the hard case.
We now assume that the results (31) and (36) apply to any hard
cylinder and that (37) and (38) apply to any soft cylinder. Of course
the cylinder must have a smooth cross section . As a first check on
these results we see that when b = (31) and (37) agree with the
results (A17a) and (AlTb) of W. Franz ^ -' for a circular cylinder.
5. Diffraction by a parabolic cylinder .
As a check on the higher order corrections to the decay exponents
and diffraction coefficients which were derived in Section 2 we now
consider the problem of diffraction by a parabolic cylinder. Our
solution will closely parallel that of S . 0. Rice L -' . However, we
find it more convenient to use parabolic cylinder functions which differ
from his and hence we will rederive his results. We again consider the
problem of evaluating the field on the surface of a hard parabolic
cylinder due to an incident plane wave. For convenience we first
consider the diffraction problem for an incident cylindrical wave and
then obtain the plane wave result by a limiting procedure.
To formulate the diffraction problem we take the z axis of an
(x,y,z) rectangular coordinate axis to be parallel to the generators
of the parabolic cylinder. In the (x,y) plane we introduce parabolic
coordinates {i,T]) through
X = ^Ti
1, 2 P. ^^)
y = 2(^ "^ ) •
1^ -
Here t| > and -oo < | < od . The parabolic cylinder is defined by
■n = constant = n . The line source is located at y = 0, x = x , i.e.,
' o o
1/2
P = n = a = X . The wave fiinction u(|,ri) then satisfies the equation
' o
u,, + u + k (I + T] )u = 5(^-a)S(Ti - a) . (2]
55 Tn
In addition u satisfies the boundary condition
u^(a,0 = 0, (5)
and the Sommerfeld radiation condition
llm r( iku - u ) =
r->oo
Wow to find u we first note that the product 0(5)t(Ti) satisfies (2)
with the delta f^lnctions replaced by zero if and \|r satisfy the ordinary
differential equations
ilf" - k^(b^ - j]^U = (5)
0" + k^(b2 + 1^)0 = 0. (6)
Here b is an arbitrary separation constant. We next note that for an
infinite set b of values of b^ there exist solutions of (5), ^^(ti),
n '■'■
which are 'outgoing ' and for which
Since the polar coordinate variable r is equal to i +r\ /2 we take the out-
going condition on i to mean that as tj -> 00 ^ \|r -> Ae ^ ' . Here and in
the following A will denote a generic amplitude function.
- 15 -
^Ja) = .
We next assume that the ^ {t\) are complete and express u as
03
u(5,Ti) = XI L(?) i'i-^)- (7)
By exactly the same calciilation as was carried out in reference
[9] it is easy to show that
CD
f i (ti)* (ti) = - 6 (2k^ )"^ 'I' (ti ) 4: t'(n ) • (8)
j n^ ' m^ ' nm^ n' n 'o db n^ 'o
Here 5 is the Kronecker delta and S/Sb \Jr (t] ) is the value of S/(^b\|r (t])
evaluated at b = b and ri = t] . Thus upon substituting (7) into (2)
miiltiplying by \lr {t\) , integrating from t] to cd , and making use of (8)
we find that ^ satisfies
^n
2 2 ? ^ -SkT) t (a)
n^ 'o' ' n^ 'o
To solve (9) we first characterize the solutions of (6) by means
of their asymptotic expansions as k — > 00 . As k -> 00 there exist
solutions 0^ ' (i) and 0^ ' (l) having the following asymptotic expansions
/^)U)-(^'.|2)-^/'exp[ik |(b2.^V/'d|]
(10)
16
<(2)
5
:0 - (t^ + l^)"^''"" expC-ik j (b^ + 5^)^/2 d|] . (11)
A simple calculation shovs that as M — > oo
0^"^^— A exp[lk| 1 5 1/2] (12)
0^^^— A exp[-ik||||/2] • (13)
It is thus apparent that as | — > oo, is the outgoing solution of (6),
(2)
while as | — >-oo , is the outgoing solution of (6). Since the variable
I takes on both positive and negative values, we see that for | > | the
solution of (9) is proportional to (l) , while for ^ < | the solution
(2)
of (9) is proportional to (l)- These conditions together with the
jump conditions imposed by the delta function allow a unique determination
of ^ (I ) . We then find that for ^ < |
^n^ ' o
oo r . 0[^\a)i (a)
u(^,n) = ik X: b^ tj 11)0^2^(1) -^ S . . (li,)
Now to pass to plane wave excitation we multiply by
17
JjtiA o5/2 1/2 ,1/2 ^ ^-ika^
c = e^"^/" 2^/^ jt-"/^ k^/^ a e-""^ (15)
and let a -> oo . In order to evaluate this limit we require the asymptotic
expans on of the function ij; (a). Using the methods of Olver ^ -' as in Section II
we find
, , , y\lh ^1/3 -1/^2 2.-1/1+ „,,l/5 -iJt/5 ,2/5 ^v (^f-.
Here
l'?!""' I (b^-.V^dn. (17)
When T] > b;, (l6) becomes
, , . irt/l2 -1/6 ^-1 -1/2, 2 ,2x-l/i+ p, \ , 2 ,2>l/2- -i ,,pN
tC^j'-^e ' k ' 2 rt (ti -b) ' exp [ik (t] - b ) ' dry. (lo;
b
Now a simple calculation shows that as n — > oo
2 2
, 2 ^2xl/2 ^ l3 1 2ti -n /,r,\
b
Also as I — > oo
2 2
2i r
b ^ ^
(,2 . b^)!/^ dl ^ ^ log ^ . 4- . (20)
Upon using (l9) in (l8) and (20) in (lO) we find
18 -
lim c0^ '(a) t (a) = 2 ' k ' e^ '
a— >oo
n n
(21)
Thus for ri = ri and for an incident plane wave (l^) becomes
o
,, . A/.3 4rti/3 5I/2 ^ ,
/(2)
(I)
^b ^^i)u^)
(22)
To obtain the asymptotic expansion of (22) as k — > 00 we proceed exactly
as in the case of the elliptic cylinder. We first find the following three
term asymptotic expansion of b from the condition that t (t] ) = 0.
b = Ti +
n 'o
'o o n 'o
Then upon using (25) in (16) and (ll) we find
sM v,„) . ,-^ ;^^-/\A(^)2i/2 ,y^ .V3 , 0(1.^)
(24)
f'(l)=(./-5^)-''"'exp
-1. I (./ . S^j^/^dS - lk^/'.„,^''' (25)
- 2 4/5 I
IT T] '
_ ( , 2 ,2.-1/2^^ ^ n ^o f , ^ .2.-5/2
30 k
X (Q^^/\ - 7)d| + TIT27T 1 ^^^ ■" ^^^'^'^^^^
n 'o o
19
Upon substituting (2^+), (25) and (23) into (22) we find
CD ( r-
u(|,^^) = ^^//^(ti/ + ?^)"^/^ Z [VK^r^ exp -ik I (. 2+|2^l/2d| (26)
n=o L /
k o
n ^O o j
In the case of the parabolic cylinder the incident rays are parallel to
the X axis and the diffracted ray to the point (|,t] ) follows the path QP as
shown in Figure 2.
Figure 1
A cross section of the elliptic cylinder
showing the points Q and Q at which two
Incident rays are tangent to it. The in-
cident field is a plane wave coming from
the right . The tangent rays produce
diffracted rays which travel distances t
and tp to a point P on the surface .
y
-\ >
Figure 2
A cross section of the parabolic
cylinder. The incident field is
a plane wave coming from the right .
The diffracted ray follows the
parabolic path QP.
20
A simple calculation shows that the element of arclength along the parabola,
ds, is given by
ds = (1^ + Ti/)^/2 |d|| ■ (27)
Thus the first term in the exponent in (26) is ikt, since | is negative.
Similarly, we find that the radius of curvature of the parabola, b, is given
by
-1,2 ,2x5/2 , ov
b = tIq i\ + i )' (28)
Upon using (27) and (28) a simple calculation shows that the exponent in
(26) can be written as
7^ exp[ikt + I P^ ds] . (29)
Here and y are defined by (51) and (52) of Section II. Upon applying
the geometric theory of reference [5] to the present case it is easy to show
that the geometric construction agrees with (26) to lowest order in k
Again we have in (26) higher order corrections to the diffraction coefficients
B (P) and the decay exponents P as given by equations (5^) and (55) of
Section II. These corrections are identical with those given by equations (51)
and (56) of Section II.
21 -
k . Integral equation method .
We will now derive the asymptotic expansion of each mode of the diffracted
field on an arbitrary convex cylinder by a different method. In this method
we begin with an integral equation and obtain a formal asymptotic solution
of it. This asymptotic solution coincides with the expression for a mode
given by the geometric theory of diffraction^ with the corrected decay
exponents and diffraction coefficients found in section 2. This independent
derivation, which follows the procedure used by W. Franz and K. Depperman ^ ^
in the case of a circular cylinder, confirms our previous result.
We consider the two dimensional problem of finding a function u(x,y)
satisfying the following equations
(V^ + k^)u =0 In D (l)
bu/hn = on C (2)
lim rl/2(|^ _ .j^^) ^ Q _ (3)
i^>oo
Here C is a given simple smooth convex curve with a piecewise continuous
second derviative. If C is closed, D denotes its exterior. If C is open
and extends to infinity, D denotes the non-convex portion of the plane,
bounded by C .
From (l) - (3) it follows that on C, u satisfies the following integral
equation
u(s) = -i j u(s')^ H^^) [kr(s,s')]ds' ^ (1,)
C
- 22
Here s denotes arclength along C measured from some fixed pointy, u(s)
is the value of u at the point s on C, r(s,s') is the distance between
the points s and s ', and the nonnal n' points into D.
If C is closed^ the only single-valued solution of (*+) '.s u h 0.
If C is open, presumably the only bounded solution is also u = 0. There-
fore if u is to represent a mode, it must be multi-valued in the former
case, or luibounded in the latter case. Consequently we assume that on C
a single mode u has i.he following asymptotic expansion for large values of k
u(s) '^ exp
oo
iks + ) V (s)k
'- — -, n
n=-l
-n/5
(5)
The coefficients v (s) are to be determined by requiring (5) to satisfy [W)
asymptotically .
Before inserting (5) into {h) , we note that for large values of k the
function m> ' rkr(s, s ' )1 /^n' has the asymptotic expansion
o *-
(6)
Sh^^^ [kr(s,s']
^ ,2k 1/2 i(kr + «A) ^ (-lf(O.ni) r^ _ m+ l/2 .
The symbol (0,m) is defined by
(O.m) = p (| + m)/m! P (| " "i) • ^^'■
Now we insert (5) and (6) into {h) and then divide by the left hand side of
the resulting equation. In this way we obtain the equation
- 23
l^e-"'^/'^(k/2n)
/\v/o.^l/2 ( ar -1/2
;, r-^/^ Yl (-ir(0,m)(2ikr)-
m=o
_ m+1/2"
Ikr
r 00 _^/
X exp [lk(r-s-s '31 exp Yl ^ ' (v (s') - v (s)
n=-l
ds'
(8)
In order to determine the v (s) from (8) we first expand the
Integral in (8) asymptotically for large values of k. We perform
this expansion by using the concept of stationary phase. The
derivative of the phase of the integrand is 1 + dr/ds', which
vanishes if dr/ds' = -1. This condition is satisfied only at s ' = s,
and then only if dr/ds ' denotes the one sided derivative computed
with s ' < s . Thus to evaluate the integral we expand the integrand
in the one sided neighborhood s ' < s of the point s' = s. For this
purpose we use the following expansions which are derived in the
appendix
= J1 <^„(s)(s-s')
n=l
-1/2 s,/^„. . _ <hl (s-s')l/2£ p^^'3)(s-s')^
00
n=o
(9)
(10)
Z. (-lA0,m)(2ikr)-
m=o
i-^1 = E e(s,k)(s-s'f (11)
ikr
J- ^-p n TTh.^ f1rqt few of the coefficients
Here 'c(s) denotes the curvature of C. The tlrsx lew 01
c n and 6 are listed in Table I.
n' ^n n
- 2k -
We now Insert (9) - (ll) Into (8), making use of the explicit
values of p , c , Cp and c . We also expand v_ (s') in a power series
about the point s' = s. Then (8) assumes the form
5jtiA / M, iQ n1/2 f / ,>l/2 r ik/c (s) f ,x3
l^e^ / '<(s)(k/8rt) / (s-s'j ' exp ok^ (s-s')^
k^/5v_^(s)(s-s')
F(k^s^s')d-s' .
(12)
The ftinctlon F(k,s,s') appearing in (l2) is defined "by
F(k,s,s') = exp
00 00 00 /^ v^ '(s)(s'-s)
n/3 n "^ ' '- '
ikx: cjs)(s-s-)" + i: TL k
- n.=h n=-l m=l
(13)
k^/^v_^(s)(s'-s)
00
YL
n=-co
YL Pjs)(s-s')^^ p Js,k)(s-s')''
In (13) V (s) denotes the m-th derivative of v (s).
To complete the asymptotic evaluation of the integral we introduce
the new variable t by means of the definition
3_3. =e-/6f^2^1^/\
,'^'"(s)k^
(IM
When (1^) Is used in (15), it shows that F(k, s^s') has an expansion of
the form
- 25
oo
F(k,s,s')^l+ Yi k'""/^ b^(t,s) . (15)
n=l
.-n/3
We next define a(s) by the equation
v_^(s) =a(s)(K(s))2/5(24)-l/5 eW6 _ (^gj
Finally ve insert (l^) - (l6) into (12)^ which becomes
l^i(3A)^/' f tl/2^-"t-t ^^^g k-"/5 Wt,s))dt. (17)
/ n=l
Upon comparing coefficients of the various powers of k in the asymptotic
form of the integral equation (17), we obtain the following set of equations
00 .,
.,,/ ^l/2 f ,1/2 -at-t^ ,, _ /_QN
1 - i(3/n) ' t ' e dt = (I8)
3
^1/2 ^-at-t' t (t,s)dt =0, n = 1, 2, ... (19)
From these equations we shall determine the coefficients v (s).
W. Franz'- -^ has shown that the left side of (18) can be rewritten
in terms of the Airy function A defined in equation (ll) of section 2.
Thus (18) becomes
- 26
iJt/5 \ / o"i'^/5
12 -W6 ,■ . <- J W. ^^ a = . (20)
Tl
k'
175 " M" -T75
The appropriate value of a is determined by the vanishing of the A'
factor in (20). If we denote by q the roots of the equation A'(q )=0,
then the values a of a are given by
a = a = -e
l«/5 1,1/3 q^ . (21)
It vill be useful to introduce the function h(a) defined by
h(a) = t
-1/2 ^-at-t5 ^^ _ (22)
Franz '- J has shown that
and that h satisfies
h'"(a) = - I h(a) - I h'(a) . ■ (2^+)
27
To determine the consequences of (l9) we must first compute the b
n
We shall calculate only b and b . To do so we substitute (ih) and (l6)
into (15), expand the exponential functions and multiply together the
resulting series in powers of k ' . In this way we obtain
-iit/6 /„, sl/3 K \ (-2 ■ A ^ a ^2 h\ . ^,
\ = e (24) -373 /_ - v^ -,j t + - t + t ^ (25)
2 8(24)^/\ <c2/5 1 ^^
X f tl_ [jL _K^, _ii.\ , ^ /:i__ 13£L_^ (26)
2.2,4 ^,,^5 / .. 4 ^^.2\ .2^6 .2 ,8
g /< t 24t / ;<:'<: k 33'<' 1 a.x t /c t
^ 13?°^ "" 7°73 r 80 ■*■ 1920 " 720 y ^ ^To73 "" ^7573
When (25) is inserted into (19) an equation for v is obtained. This
equation contains integrals of the form
.3
j t^-1/2 e-^^-* dt = (-l)V^)(a).
(27)
In (27) the Integral has been expressed in terms of the n-th derivative of
h(LL) which is defined by (22). Thus from (19) we find that the right hand
side of (25) must vanish when t is replaced by (-1) h^''^ (a). This yields
(l-u) -
28
I h'"(a) - h^^^(a) = 0. (28)
By using (2^1) we find that
,(IV), V _ 1 V,, _ 2: ^,.
h^^'^(a) = -ih' -^h" (29)
h^V\a) =f8h+|- h' -|h" (30)
When (2^) and (30) are used in (28), the following expression for v results
o 6 k
(51)
Upon integrating (31) we finally obtain for v the expression
V = log ■<-' + 5 . (32)
o
Here 5 is an Integration constant.
The analysis of (19) for "the case n = 2 proceeds in exactly the same
way. In this case we obtain the condition that the right hand side of (26)
vanishes when t is replaced by (-1) h^ '(a) . In order to simplify
the resulting expression we must express the sixth through ninth derivatives
of h In terms of h, h', and h" . Upon doing this we find
t *
In order to avoid writing cumbersome equations we denote by b the right
hand side of (26) with t^ replaced by (-1)""^"^ h^"'^"^^(a).
- 29
,(VIII)._|,.I..2^,,|5,„ .|!,., (35)
We also note that
h"(a) = - I h(a). (37)
This result follows upon differentiating (23) twice^ then using the
equation satisfied by A(x)
A + I A = 0, (38)
and finally noting that when a is defined by (2l)
Al^) .0. (59)
We now insert the preceding relations for the derivatives of h,
together with (31) and (37) into b . We then find that the coefficient
of h' vanishes and that the equation bp = may be solved to yield
50
y^ - 6(24)1/5 .2/3 ^-1 ^-ln/6 I
.2/3
1 1 < ]^ 2 1 K
ZJK \6k
K 1^ K^ .\ Ja? ^ 24 (■]_ a?
2 \TTJ5 ^ 36 "VT/ "■ 648 ~io75 ■" :T573 U^ ' 3^
K
(^0)
X ^- '^
4
33 .2
1920 720
ko? k^ 1 K-2 f 293£ _ 91_
81 ^10/3 "^ 2 ^10/3 I324 " 216
We next make use of (2l) of this section and (21) of section 2 and set
K = b = the radius of curvature of C. Then a straightforward calculation
shows that (4o) may be written as
iT
^1 = 3^ ^.■^/n^.-vn^-
-1/5
201
i I.. -4/3 ^ I ^-4/3 ^2 _ ^
1/3
s 5 ss
(^1)
Let us now combine our results (l6)^ (21), (32), and (4l). By
Inserting them into (5) we obtain the asymptotic expansion of u(s)
-1/3
on the cylinder C up to and including terms in k
s
u '^ E b ' ( s ) exp
iks + i k^/^ T^ I b"^/^(s)ds
-2/5.
^^"^' .^ { fb-V3 , 16 ,-4/5,2 _ 8b^ ,
9 s 5 £
30
=)
ds
ikll/5
20t
n i V 9 s 5 ssy
ds + . . .
.(^2)
- 31 -
Here E denotes an arbitrary constant.
Let us now compare the result (^2) vith the expression for a mode
given by the geometric theory of diffraction [5] • That theory yields
for u a single term of the sum in (33) of section 2. Let us insert into
that equation the improved decay exponents (31) and diffraction coefficients
(36). Then we find that each term of (33) coincides with (^2) provided
that the product of the constant coefficients in (33) is equated to the
constant E in (^2). This agreement between the two methods of obtaining
the improved decay exponents and diffraction coefficients again confirms
the results of section 2. The method of the present section can also be
modified to apply to soft cylinders, on which u = 0.
- 52 -
Appendix
In order to calculate the quantity r(s,s') in the neighborhood of
s = s ' we first ohserve that if x(s) is the position vector to the curve
C, then
2
r =
.2
(x(s) - x(s-)) . (i;
By Taylor ' s theorem
oo ■'(n)/ X
x(s')-x(s)=^ {s<- sf ^ /" ' ' • (2)
n=l
Thus, upon taking the dot product of (2) with itself we find
? = }_ (s'- s) b .
P=2
Here
p- 1 ^(n)/ V -^Cp-n),
Y ^( s)- X I
n: (p-n)l
. = E ^'"'[s)-,^':::'i^) . (^
n=l
Since s is arclength along C we have
x(s). x(s) =1, _ (5)
and
x(s).x(s) = 0. (6)
- 53 -
From the Frenet equations of differential geometry we have
x(s) = KK X - K X ' (7)
Upon using (5), (6), and (7) recursively to obtain the higher derivatives
of x(s) in terms of x(s) and x(s) we see that b can be expressed in terms
of K and its derivatives . In this way we find
(8)
2 . h
- K ~ KK - 1 2 1 K
b^ = 1; b^ = 0; b^ = — ; b^ = ^ J ^6 = T5 ^ -To^'^-'Wo
Then upon taking the square root of the right hand side of (3) we find
= Z «^(S-B')^ . (9)
n=l
Here
2
"bi, -'be b^ b],
„ _v. „ _v. c - -^ ' c --5- c - (— - — ) (10)
c^ - b^, ^2' ^y ^3 ~ 2 > % ' 2 ' ""5 " ^2 8 '' •
From (8) and (lO) the entries for c in Table I are obtained.
-1/2
In order to calculate r ' Sr/Sn ' we note that the unit normal to
C at s' is K (s') x(s') = V (s') and hence
l^='(?)-V-)=^ . (11)
Thus by making use of (2) and the Taylor expansion of v (s') about s = s'
we obtain
- 3^
Sr_ _^ r.._ .^P . (12)
p=l ^
Here
f =£ ^, ^ .^ • (13)
p ^3_ k: (p-k):
Again upon using the Frenet equations recursively the coefficients f
may be easily evaluated to obtain
5
f^ = 0; ^2 ^ ' 2 ' ^3 ^ ~ 3 ' ^^ ^ ~ Q'^ 2^ ■
Now by applying the binomial theorem to (3) we find
where
(Ik)
^-5/2 = (s- s')-5/2 (bg - f b^(s-s')^ + I b^(s-s')5 + ...) . (15)
Thus upon multiplying (15) and (l2) we find
^-1/2 ^, . - 4^ (s-s.)'/" £ Pjs)(s-s.)- , (16)
^^ 2f^ . , ,
Thus from (8)^ (l^)^ and (17) we obtain the values of p given in Table I.
Upon using (9) and the values of (0,m) given in section ^ it is a simple
matter to calculate the values of B and 6^ as given In Table I and to
o -1
conclude that P (s-s')" = o (k"^/^) f^-^ n ,/ 0, -1 and s-s ' = 0(k"'^'^)-
- 3$ -
Table I
-1
5i/8k
2 K
'3 K
K
kK
K K
12 "^ 16
2l^
KK
2^
.2 .. 1|
K KK K
90 " 80 "^ 1920
- 36 -
References
[l] Watson^ G.
[2] Bremmer, H.
[jj Franz, W. and Depperman, K.
[h] Keller, J. B.
[5] Keller, J. B.
[6] Franz, W.
[7] Levy, B. R., and Keller, J.B.
[8] Rice, S. 0.,
[9] Levy, B.R.
[10] Kazarinoff, N. D. and Ritt, R. K.
"The diiTraction of electric waves
by the earth and the transmission of
electric waves around the earth ' ,
Proc. Roy. Soc . (London), Vol. A95,
pp. 83-99, Oct. 1918, pp. 5^6-565,
July 1919.
'Terrestrial Radio Waves', Elsevier
Publishing Co., New York. N.Y.
'Theorie der Beugung am Zylinder unter
Berucksichtigung der Kriechwelle ' , Ann.
der Phys., Vol. 10, Wo. 6. pp. 361-375^
1952.
'A geometric theory of diffraction'.
Calculus of Variations and its
Applications, Proc. of Symposia in
Applied Math., Vol. VIII, pp. 27-52,
McGraw-Hill, New York, N.Y., 1958.
'Diffraction by a convex cylinder',
I . R . E . Trans . on Ant . and Prop . ,
Symp. on Electromagnetic Wave Theory,
Vol. AP-4, pp. 312-321, July 1956.
'Uber die Greenschen Fujiktionen
des Zylinders and der Kugel',
Zelt. fur Naturf ., Vol. 9a,
pp. 705-716, 195^.
'Diffraction by a smooth object',
Comm. Pure and Appl. Math., 12,
No. 1, 159-209, 1959-
'Diffraction of plane radio waves
by a parabolic cylinder'. Bell Sys .
Tech. J., Vol. 33, PP- ^17-502,
March 195^+.
'Diffraction by an elliptic cylinder'.
New York Univer., Inst, of Math. Sci.,
Div. of EM Res., Res. Report Wo. EM-121,
(1958).
'Scalar Diffraction by an elliptic
cylinder', Univer. of Michigan,
Res. Inst., Scientific report.
[ll] Levy, B. R. and Keller, J. B.
[12] Ritt, R. K. and Kazarinoff, N.
[13] Olver, F. W. J.
'Diffraction by a spheroid'. New York
Univer., Inst, of Math. Sci., Div . of
EM Res., Res. Report No. EM-130(l959
'Studies in radar cross sections XXX,
The theory of scalar diffraction with
application to the prolate spheroid',
Univer. of Michigan Res. Inst.,
Scientific Report No. h, I958.
'The asymptotic solution of linear
differential equations of the second
order for large values of a parameter',
Phil. Trans. Roy. Soc. of London,
Series A, 2kj_, pp. 307-368(1954.
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